General Observation on Prime Numbers

Date: 09/03/2004 at 19:31:48
From: Stanley
Subject: prime numbers
Is it true that all prime numbers greater than 5 are of the form
6n + 1 or 6n - 1? I read this on a website, but it's hard to believe.

Date: 09/05/2004 at 15:02:28
From: Doctor Beryllium
Subject: Re: prime numbers
Hi Stanley,
Yes, it is the case that all primes greater than 6 are either of the
form 6k + 1 or 6k - 1.
Consider that we can divide the set of integers into the following
classes:
class 0 6k - 1
class 1 6k
class 2 6k + 1
class 3 6k + 2
class 4 6k + 3
class 5 6k + 4
class 6 6k + 5
Further notice that the the difference between integers of class 0 and
class 6 is 6 (this is because -1 is congruent to 5 modulo 6). That
means that this cycle of integer classes will repeat over and over as
we count through the integers with increasing values of k.
Now integers of class 1 have 6 as a factor. Integers of class 3 have
2 as a factor. Integers of class 4 have 3 as a factor. Integers of
class 5 have 2 as a factor. This leaves us with the fact that all
primes must either be from class 2 or class 0 or class 6. But we know
integers from class 6 are the same integers as from class 0.
This shows that all primes must be of one of the following forms:
6k - 1 == 6k + 5 (congruent modulo 6)
or
6k + 1
- Doctor Beryllium, The Math Forum
http://mathforum.org/dr.math/